We also emphasize that this formal expansion of course does not have to be convergent. Instead, the numerical coefficients may grow rapidly with increasing orders in the expansions. In fact, expansions in the number of derivatives and powers of the field are typically non-convergent but asymptotic [1]}.
Besides, we note that by construction a perturbative expansion in the field strength is of course insensitive to manifestly nonperturbative phenomena, such as the principle possibility of electron-positron pair-production via the Schwinger effect [2]}, [3]}, [4]}. These effects can, however, be safely neglected in the parameter regime to be considered below; cf., e.g., Refs. [5]}, [6]}, [7]}, [1]}, [9]} and references therein.
